Derivative corrections to the Heisenberg-Euler effective action

We show that the leading derivative corrections to the Heisenberg-Euler effective action can be determined efficiently from the vacuum polarization tensor evaluated in a homogeneous constant background field. After deriving the explicit parameter-integral representation for the leading derivative corrections in generic electromagnetic fields at one loop, we specialize to the cases of magnetic- and electric-like field configurations characterized by the vanishing of one of the secular invariants of the electromagnetic field. In these cases, closed-form results and the associated all-orders weak- and strong-field expansions can be worked out. One immediate application is the leading derivative correction to the renowned Schwinger-formula describing the decay of the quantum vacuum via electron-positron pair production in slowly-varying electric fields.

Automated summary

The article demonstrates the efficient determination of leading derivative corrections to the Heisenberg-Euler effective action from the vacuum polarization tensor evaluated in a homogeneous constant background field. With an explicit parameter-integral representation derived for these corrections in generic electromagnetic fields at one loop, the focus then shifts to magnetic- and electric-like field configurations where closed-form results and all-orders weak- and strong-field expansions can be obtained. An important application highlighted is the leading derivative correction to the well-known Schwinger formula, describing the decay of quantum vacuum through electron-positron pair production in slowly varying electric fields.